Fourier Transforms in Maple - Maple Help

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Fourier Transforms in Maple

 Fourier transforms in Maple can be categorized as either transforms on expressions or transforms on signal data.
 To compute the Fourier transform of an expression, use the inttrans[fourier] command. For more details on this command, see the inttrans[fourier] help page.
 To compute the Fourier transform of signal data, the following commands are available:
 • SignalProcessing[DFT] : This command computes the discrete Fourier transform of an Array of signal data points. The SignalProcessing[DFT] command works for any size Array. For more information, see SignalProcessing[DFT].
 • SignalProcessing[FFT] : Similar to the SignalProcessing[DFT] command, SignalProcessing[FFT] computes the discrete Fourier transform of an Array of signal data points. The difference between the two commands is that the SignalProcessing[FFT] command uses the fast Fourier transform algorithm. Note: SignalProcessing[FFT] requires that the size of the Array must be a power of 2, greater than 2. If the Array passed to SignalProcessing[FFT] does not meet this requirement, the SignalProcessing[DFT] command is used instead. Similarly, SignalProcessing[InverseFFT] calls SignalProcessing[InverseDFT] when the passed Array does not meet this requirement. For more information, see SignalProcessing[FFT].
 • DiscreteTransforms[FourierTransform] : The DiscreteTransforms[FourierTransform] provides similar functionality to that of SignalProcessing[DFT]. There are some options available in DiscreteTransforms[FourierTransform], such as padding, that are not available in SignalProcessing[DFT]. For more information, see DiscreteTransforms[FourierTransform].
 Note: Typically, SignalProcessing[DFT] and SignalProcessing[FFT] are slightly more efficient than DiscreteTransforms[FourierTransform].
 The table below provides a summarized comparison of the discrete Fourier transform commands mentioned above.

 Feature SignalProcessing[FFT] SignalProcessing[DFT] DiscreteTransforms[Fourier Transform] input single rtable yes yes yes input two rtables (Re/Im) yes yes yes higher-dimensional transforms yes yes yes specify single dim for higher-dimensional transforms no no yes output single rtable yes yes yes output two rtables (Re/Im) yes yes yes padding no no yes apply transform only to initial segment no no yes in place yes yes yes specify output rtable yes yes no specify working storage no no yes size of Array: power of 2 yes yes yes size of Array: other yes (dispatch to DFT) yes yes

Examples

 > $\mathrm{signal}≔\mathrm{sin}\left(t\right)\mathrm{exp}\left(-\frac{{t}^{2}}{100}\right)$
 ${\mathrm{signal}}{≔}{\mathrm{sin}}{}\left({t}\right){}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{100}}}$ (1)
 > $\mathrm{plot}\left(\mathrm{signal},t=-30..30\right)$
 > $\mathrm{transform}≔\mathrm{inttrans}\left[\mathrm{fourier}\right]\left(\mathrm{signal},t,s\right)$
 ${\mathrm{transform}}{≔}{-}{10}{}{I}{}\sqrt{{\mathrm{\pi }}}{}{\mathrm{sinh}}{}\left({50}{}{s}\right){}{{ⅇ}}^{{-}{25}{}{{s}}^{{2}}{-}{25}}$ (2)

The transform is purely imaginary:

 > $\mathrm{evalc}\left(\mathrm{\Re }\left(\mathrm{transform}\right)\right)$
 ${0}$ (3)

This is what the imaginary part looks like:

 > $\mathrm{plot}\left(\mathrm{\Im }\left(\mathrm{transform}\right),s=-3..3\right)$
 > $\mathrm{inttrans}\left[\mathrm{invfourier}\right]\left(\mathrm{transform},s,t\right)$
 ${\mathrm{sin}}{}\left({t}\right){}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{100}}}$ (4)

Turn the original signal into data by sampling:

 > $\mathrm{data}≔\mathrm{Array}\left(1..80,i↦\mathrm{evalf}\left(\mathrm{eval}\left(\mathrm{signal},t=\frac{i}{4}\right)\right)\right)$
 ${\mathrm{data}}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.2472493801}& {0.4782284717}& {0.6778153055}& {0.8330982086}& {0.9342719766}& {0.9753019572}& {0.9543081382}& {0.8736433648}& {0.7396636850}& {0.5622125499}& {0.3538622664}& {0.1289739763}& {-0.09734987013}& {-0.3103399929}& {-0.4965810793}& {-0.6449045459}& {-0.7470908727}& {-0.7983356335}& {-0.7974529355}& {-0.7468109761}& {-0.6520150601}& {-0.5213720678}& {-0.3651855797}& {-0.1949415782}& {-0.02245018091}& {0.1409909856}& {0.2853511479}& {0.4024873310}& {0.4865936932}& {0.5344562763}& {0.5455063352}& {0.5216811081}& {0.4671149321}& {0.3876949535}& {0.2905236192}& {0.1833342324}& {0.07390606951}& {-0.03047787547}& {-0.1234939721}& {-0.2001341823}& {-0.2569380786}& {-0.2920941320}& {-0.3054141956}& {-0.2981943590}& {-0.2729827805}& {-0.2332802508}& {-0.1832019433}& {-0.1271290188}& {-0.06937673750}& {-0.01390182491}& {0.03593343608}& {0.07752902163}& {0.1091418791}& {0.1299085975}& {0.1398024388}& {0.1395353879}& {0.1304188346}& {0.1141980569}& {0.09287587668}& {0.06853983412}& {0.04320521664}& {0.01868354009}& {-0.003517056003}& {-0.02225629117}& {-0.03679098198}& {-0.04677160933}& {-0.05221092953}& {-0.05343083141}& {-0.05099444295}& {-0.04563063549}& {-0.03815763932}& {-0.02941158576}& {-0.02018456742}& {-0.01117540149}& {-0.002954832805}& {0.004054455326}& {0.009583845211}& {0.01351257059}& {0.01585170961}& {0.01672117554}\end{array}\right]$ (5)
 > $\mathrm{SignalProcessing}:-\mathrm{SignalPlot}\left(\mathrm{data}\right)$
 > $\mathrm{tdata}≔\mathrm{SignalProcessing}:-\mathrm{DFT}\left(\mathrm{data}\right)$
 ${\mathrm{tdata}}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.449765990214555}{+}{0.}{}{I}& {0.505210797657416}{+}{0.0390663687494100}{}{I}& {0.871158772341445}{+}{0.0744865266272709}{}{I}& {1.11827605677666}{-}{1.60320578988416}{}{I}& {-0.833974901255164}{-}{0.664082006094420}{}{I}& {-0.315964580561808}{-}{0.123878845533608}{}{I}& {-0.166630985580569}{-}{0.0788471078261399}{}{I}& {-0.105352012138145}{-}{0.0595621316126666}{}{I}& {-0.0724092958555620}{-}{0.0482794303311701}{}{I}& {-0.0522288158121224}{-}{0.0407062195325426}{}{I}& {-0.0388242915684034}{-}{0.0351920466279138}{}{I}& {-0.0294082117174462}{-}{0.0309536152571132}{}{I}& {-0.0225148313113225}{-}{0.0275667649429405}{}{I}& {-0.0173047300967625}{-}{0.0247798253438997}{}{I}& {-0.0132655064949832}{-}{0.0224330666732571}{}{I}& {-0.0100684268175711}{-}{0.0204197230546756}{}{I}& {-0.00749398324012806}{-}{0.0186654357259765}{}{I}& {-0.00539069323582706}{-}{0.0171166540582818}{}{I}& {-0.00365108985686116}{-}{0.0157337252108298}{}{I}& {-0.00219711705216668}{-}{0.0144865880008114}{}{I}& {-0.000970919849976031}{-}{0.0133519876984611}{}{I}& {0.0000711465796272747}{-}{0.0123116147660412}{}{I}& {0.000962519122718898}{-}{0.0113508269453421}{}{I}& {0.00172916454491457}{-}{0.0104577505722189}{}{I}& {0.00239149118681304}{-}{0.00962263465913052}{}{I}& {0.00296570883192298}{-}{0.00883737750161765}{}{I}& {0.00346481191751634}{-}{0.00809517494757373}{}{I}& {0.00389930099312579}{-}{0.00739025361514114}{}{I}& {0.00427771919553845}{-}{0.00671766684027044}{}{I}& {0.00460705573117073}{-}{0.00607313531064888}{}{I}& {0.00489305257703095}{-}{0.00545292250998158}{}{I}& {0.00514043989624955}{-}{0.00485373469997701}{}{I}& {0.00535311762499700}{-}{0.00427264091457455}{}{I}& {0.00553429690919023}{-}{0.00370700693220772}{}{I}& {0.00568661018298807}{-}{0.00315444171561106}{}{I}& {0.00581219727669599}{-}{0.00261275145103020}{}{I}& {0.00591277217762966}{-}{0.00207990135986910}{}{I}& {0.00598967510525535}{-}{0.00155398186211123}{}{I}& {0.00604391080315031}{-}{0.00103318002042891}{}{I}& {0.00607617818568547}{-}{0.000515752070609053}{}{I}& {0.00608688863272777}{-}{7.75791922889773}{×}{{10}}^{{-18}}{}{I}& {0.00607617818568548}{+}{0.000515752070609030}{}{I}& {0.00604391080315033}{+}{0.00103318002042890}{}{I}& {0.00598967510525537}{+}{0.00155398186211121}{}{I}& {0.00591277217762971}{+}{0.00207990135986899}{}{I}& {0.00581219727669578}{+}{0.00261275145103019}{}{I}& {0.00568661018298803}{+}{0.00315444171561109}{}{I}& {0.00553429690919016}{+}{0.00370700693220771}{}{I}& {0.00535311762499700}{+}{0.00427264091457455}{}{I}& {0.00514043989624961}{+}{0.00485373469997699}{}{I}& {0.00489305257703101}{+}{0.00545292250998158}{}{I}& {0.00460705573117094}{+}{0.00607313531064867}{}{I}& {0.00427771919553838}{+}{0.00671766684027042}{}{I}& {0.00389930099312576}{+}{0.00739025361514109}{}{I}& {0.00346481191751632}{+}{0.00809517494757372}{}{I}& {0.00296570883192299}{+}{0.00883737750161764}{}{I}& {0.00239149118681302}{+}{0.00962263465913052}{}{I}& {0.00172916454491458}{+}{0.0104577505722189}{}{I}& {0.000962519122718887}{+}{0.0113508269453421}{}{I}& {0.0000711465796273081}{+}{0.0123116147660412}{}{I}& {-0.000970919849976022}{+}{0.0133519876984611}{}{I}& {-0.00219711705216667}{+}{0.0144865880008114}{}{I}& {-0.00365108985686119}{+}{0.0157337252108298}{}{I}& {-0.00539069323582699}{+}{0.0171166540582817}{}{I}& {-0.00749398324012806}{+}{0.0186654357259765}{}{I}& {-0.0100684268175711}{+}{0.0204197230546757}{}{I}& {-0.0132655064949831}{+}{0.0224330666732570}{}{I}& {-0.0173047300967625}{+}{0.0247798253438997}{}{I}& {-0.0225148313113226}{+}{0.0275667649429405}{}{I}& {-0.0294082117174462}{+}{0.0309536152571133}{}{I}& {-0.0388242915684034}{+}{0.0351920466279138}{}{I}& {-0.0522288158121224}{+}{0.0407062195325426}{}{I}& {-0.0724092958555620}{+}{0.0482794303311701}{}{I}& {-0.105352012138145}{+}{0.0595621316126666}{}{I}& {-0.166630985580569}{+}{0.0788471078261399}{}{I}& {-0.315964580561808}{+}{0.123878845533608}{}{I}& {-0.833974901255164}{+}{0.664082006094420}{}{I}& {1.11827605677666}{+}{1.60320578988416}{}{I}& {0.871158772341445}{-}{0.0744865266272709}{}{I}& {0.505210797657416}{-}{0.0390663687494100}{}{I}\end{array}\right]$ (6)

The following calls the FFT command, which in turn calls the DFT command (since the size of the data is not a power of 2):

 > $\mathrm{tdata2}≔\mathrm{SignalProcessing}:-\mathrm{FFT}\left(\mathrm{data}\right)$
 ${\mathrm{tdata2}}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.449765990214555}{+}{0.}{}{I}& {0.505210797657416}{+}{0.0390663687494100}{}{I}& {0.871158772341445}{+}{0.0744865266272709}{}{I}& {1.11827605677666}{-}{1.60320578988416}{}{I}& {-0.833974901255164}{-}{0.664082006094420}{}{I}& {-0.315964580561808}{-}{0.123878845533608}{}{I}& {-0.166630985580569}{-}{0.0788471078261399}{}{I}& {-0.105352012138145}{-}{0.0595621316126666}{}{I}& {-0.0724092958555620}{-}{0.0482794303311701}{}{I}& {-0.0522288158121224}{-}{0.0407062195325426}{}{I}& {-0.0388242915684034}{-}{0.0351920466279138}{}{I}& {-0.0294082117174462}{-}{0.0309536152571132}{}{I}& {-0.0225148313113225}{-}{0.0275667649429405}{}{I}& {-0.0173047300967625}{-}{0.0247798253438997}{}{I}& {-0.0132655064949832}{-}{0.0224330666732571}{}{I}& {-0.0100684268175711}{-}{0.0204197230546756}{}{I}& {-0.00749398324012806}{-}{0.0186654357259765}{}{I}& {-0.00539069323582706}{-}{0.0171166540582818}{}{I}& {-0.00365108985686116}{-}{0.0157337252108298}{}{I}& {-0.00219711705216668}{-}{0.0144865880008114}{}{I}& {-0.000970919849976031}{-}{0.0133519876984611}{}{I}& {0.0000711465796272747}{-}{0.0123116147660412}{}{I}& {0.000962519122718898}{-}{0.0113508269453421}{}{I}& {0.00172916454491457}{-}{0.0104577505722189}{}{I}& {0.00239149118681304}{-}{0.00962263465913052}{}{I}& {0.00296570883192298}{-}{0.00883737750161765}{}{I}& {0.00346481191751634}{-}{0.00809517494757373}{}{I}& {0.00389930099312579}{-}{0.00739025361514114}{}{I}& {0.00427771919553845}{-}{0.00671766684027044}{}{I}& {0.00460705573117073}{-}{0.00607313531064888}{}{I}& {0.00489305257703095}{-}{0.00545292250998158}{}{I}& {0.00514043989624955}{-}{0.00485373469997701}{}{I}& {0.00535311762499700}{-}{0.00427264091457455}{}{I}& {0.00553429690919023}{-}{0.00370700693220772}{}{I}& {0.00568661018298807}{-}{0.00315444171561106}{}{I}& {0.00581219727669599}{-}{0.00261275145103020}{}{I}& {0.00591277217762966}{-}{0.00207990135986910}{}{I}& {0.00598967510525535}{-}{0.00155398186211123}{}{I}& {0.00604391080315031}{-}{0.00103318002042891}{}{I}& {0.00607617818568547}{-}{0.000515752070609053}{}{I}& {0.00608688863272777}{-}{7.75791922889773}{×}{{10}}^{{-18}}{}{I}& {0.00607617818568548}{+}{0.000515752070609030}{}{I}& {0.00604391080315033}{+}{0.00103318002042890}{}{I}& {0.00598967510525537}{+}{0.00155398186211121}{}{I}& {0.00591277217762971}{+}{0.00207990135986899}{}{I}& {0.00581219727669578}{+}{0.00261275145103019}{}{I}& {0.00568661018298803}{+}{0.00315444171561109}{}{I}& {0.00553429690919016}{+}{0.00370700693220771}{}{I}& {0.00535311762499700}{+}{0.00427264091457455}{}{I}& {0.00514043989624961}{+}{0.00485373469997699}{}{I}& {0.00489305257703101}{+}{0.00545292250998158}{}{I}& {0.00460705573117094}{+}{0.00607313531064867}{}{I}& {0.00427771919553838}{+}{0.00671766684027042}{}{I}& {0.00389930099312576}{+}{0.00739025361514109}{}{I}& {0.00346481191751632}{+}{0.00809517494757372}{}{I}& {0.00296570883192299}{+}{0.00883737750161764}{}{I}& {0.00239149118681302}{+}{0.00962263465913052}{}{I}& {0.00172916454491458}{+}{0.0104577505722189}{}{I}& {0.000962519122718887}{+}{0.0113508269453421}{}{I}& {0.0000711465796273081}{+}{0.0123116147660412}{}{I}& {-0.000970919849976022}{+}{0.0133519876984611}{}{I}& {-0.00219711705216667}{+}{0.0144865880008114}{}{I}& {-0.00365108985686119}{+}{0.0157337252108298}{}{I}& {-0.00539069323582699}{+}{0.0171166540582817}{}{I}& {-0.00749398324012806}{+}{0.0186654357259765}{}{I}& {-0.0100684268175711}{+}{0.0204197230546757}{}{I}& {-0.0132655064949831}{+}{0.0224330666732570}{}{I}& {-0.0173047300967625}{+}{0.0247798253438997}{}{I}& {-0.0225148313113226}{+}{0.0275667649429405}{}{I}& {-0.0294082117174462}{+}{0.0309536152571133}{}{I}& {-0.0388242915684034}{+}{0.0351920466279138}{}{I}& {-0.0522288158121224}{+}{0.0407062195325426}{}{I}& {-0.0724092958555620}{+}{0.0482794303311701}{}{I}& {-0.105352012138145}{+}{0.0595621316126666}{}{I}& {-0.166630985580569}{+}{0.0788471078261399}{}{I}& {-0.315964580561808}{+}{0.123878845533608}{}{I}& {-0.833974901255164}{+}{0.664082006094420}{}{I}& {1.11827605677666}{+}{1.60320578988416}{}{I}& {0.871158772341445}{-}{0.0744865266272709}{}{I}& {0.505210797657416}{-}{0.0390663687494100}{}{I}\end{array}\right]$ (7)

tdata and tdata2 are the same, up to tiny float inaccuracies:

 > $\mathrm{verify}\left(\mathrm{tdata},\mathrm{tdata2},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{tdata3}≔\mathrm{DiscreteTransforms}:-\mathrm{FourierTransform}\left(\mathrm{data}\right)$
 ${\mathrm{tdata3}}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.449765990214555}{+}{0.}{}{I}& {0.505210797657416}{+}{0.0390663687494100}{}{I}& {0.871158772341445}{+}{0.0744865266272709}{}{I}& {1.11827605677666}{-}{1.60320578988416}{}{I}& {-0.833974901255164}{-}{0.664082006094420}{}{I}& {-0.315964580561808}{-}{0.123878845533608}{}{I}& {-0.166630985580569}{-}{0.0788471078261399}{}{I}& {-0.105352012138145}{-}{0.0595621316126666}{}{I}& {-0.0724092958555620}{-}{0.0482794303311701}{}{I}& {-0.0522288158121224}{-}{0.0407062195325426}{}{I}& {-0.0388242915684034}{-}{0.0351920466279138}{}{I}& {-0.0294082117174461}{-}{0.0309536152571132}{}{I}& {-0.0225148313113225}{-}{0.0275667649429405}{}{I}& {-0.0173047300967625}{-}{0.0247798253438998}{}{I}& {-0.0132655064949832}{-}{0.0224330666732571}{}{I}& {-0.0100684268175711}{-}{0.0204197230546756}{}{I}& {-0.00749398324012806}{-}{0.0186654357259765}{}{I}& {-0.00539069323582706}{-}{0.0171166540582817}{}{I}& {-0.00365108985686117}{-}{0.0157337252108298}{}{I}& {-0.00219711705216673}{-}{0.0144865880008114}{}{I}& {-0.000970919849975982}{-}{0.0133519876984611}{}{I}& {0.0000711465796273399}{-}{0.0123116147660412}{}{I}& {0.000962519122718908}{-}{0.0113508269453421}{}{I}& {0.00172916454491455}{-}{0.0104577505722189}{}{I}& {0.00239149118681304}{-}{0.00962263465913052}{}{I}& {0.00296570883192299}{-}{0.00883737750161767}{}{I}& {0.00346481191751632}{-}{0.00809517494757373}{}{I}& {0.00389930099312586}{-}{0.00739025361514106}{}{I}& {0.00427771919553855}{-}{0.00671766684027049}{}{I}& {0.00460705573117093}{-}{0.00607313531064888}{}{I}& {0.00489305257703100}{-}{0.00545292250998161}{}{I}& {0.00514043989624960}{-}{0.00485373469997705}{}{I}& {0.00535311762499700}{-}{0.00427264091457455}{}{I}& {0.00553429690919028}{-}{0.00370700693220770}{}{I}& {0.00568661018298809}{-}{0.00315444171561108}{}{I}& {0.00581219727669591}{-}{0.00261275145103011}{}{I}& {0.00591277217762964}{-}{0.00207990135986909}{}{I}& {0.00598967510525535}{-}{0.00155398186211123}{}{I}& {0.00604391080315030}{-}{0.00103318002042890}{}{I}& {0.00607617818568545}{-}{0.000515752070608990}{}{I}& {0.00608688863272777}{-}{7.75791922889773}{×}{{10}}^{{-18}}{}{I}& {0.00607617818568547}{+}{0.000515752070609055}{}{I}& {0.00604391080315032}{+}{0.00103318002042891}{}{I}& {0.00598967510525534}{+}{0.00155398186211128}{}{I}& {0.00591277217762972}{+}{0.00207990135986902}{}{I}& {0.00581219727669581}{+}{0.00261275145103011}{}{I}& {0.00568661018298806}{+}{0.00315444171561106}{}{I}& {0.00553429690919015}{+}{0.00370700693220769}{}{I}& {0.00535311762499700}{+}{0.00427264091457455}{}{I}& {0.00514043989624960}{+}{0.00485373469997699}{}{I}& {0.00489305257703106}{+}{0.00545292250998159}{}{I}& {0.00460705573117092}{+}{0.00607313531064873}{}{I}& {0.00427771919553838}{+}{0.00671766684027043}{}{I}& {0.00389930099312571}{+}{0.00739025361514111}{}{I}& {0.00346481191751630}{+}{0.00809517494757374}{}{I}& {0.00296570883192298}{+}{0.00883737750161764}{}{I}& {0.00239149118681302}{+}{0.00962263465913053}{}{I}& {0.00172916454491457}{+}{0.0104577505722189}{}{I}& {0.000962519122718887}{+}{0.0113508269453421}{}{I}& {0.0000711465796272033}{+}{0.0123116147660412}{}{I}& {-0.000970919849976014}{+}{0.0133519876984611}{}{I}& {-0.00219711705216666}{+}{0.0144865880008114}{}{I}& {-0.00365108985686119}{+}{0.0157337252108297}{}{I}& {-0.00539069323582697}{+}{0.0171166540582817}{}{I}& {-0.00749398324012806}{+}{0.0186654357259765}{}{I}& {-0.0100684268175711}{+}{0.0204197230546756}{}{I}& {-0.0132655064949831}{+}{0.0224330666732571}{}{I}& {-0.0173047300967624}{+}{0.0247798253438996}{}{I}& {-0.0225148313113226}{+}{0.0275667649429405}{}{I}& {-0.0294082117174463}{+}{0.0309536152571133}{}{I}& {-0.0388242915684034}{+}{0.0351920466279138}{}{I}& {-0.0522288158121224}{+}{0.0407062195325426}{}{I}& {-0.0724092958555620}{+}{0.0482794303311701}{}{I}& {-0.105352012138145}{+}{0.0595621316126666}{}{I}& {-0.166630985580569}{+}{0.0788471078261399}{}{I}& {-0.315964580561808}{+}{0.123878845533608}{}{I}& {-0.833974901255164}{+}{0.664082006094420}{}{I}& {1.11827605677666}{+}{1.60320578988416}{}{I}& {0.871158772341445}{-}{0.0744865266272709}{}{I}& {0.505210797657416}{-}{0.0390663687494100}{}{I}\end{array}\right]$ (9)

Moreover, tdata and tdata3 are the same, up to tiny float inaccuracies:

 > $\mathrm{verify}\left(\mathrm{tdata},\mathrm{tdata3},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{SignalProcessing}:-\mathrm{SignalPlot}\left({⟨\mathrm{~}\left[\mathrm{\Re }\right]\left(\mathrm{tdata}\right),\mathrm{~}\left[\mathrm{\Im }\right]\left(\mathrm{tdata}\right)⟩}^{\mathrm{%T}},'\mathrm{compactplot}'\right)$

 > $\mathrm{original}≔\mathrm{SignalProcessing}:-\mathrm{InverseFFT}\left(\mathrm{tdata}\right)$
 ${\mathrm{original}}{≔}\left[\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}{0.247249380100000}{-}{5.78880680703867}{×}{{10}}^{{-17}}{}{I}& {0.478228471700000}{-}{4.63068439595559}{×}{{10}}^{{-18}}{}{I}& {0.677815305500000}{-}{8.81319038029480}{×}{{10}}^{{-17}}{}{I}& {0.833098208600000}{-}{6.51485433272886}{×}{{10}}^{{-17}}{}{I}& {0.934271976600000}{-}{3.84932699981424}{×}{{10}}^{{-17}}{}{I}& {0.975301957200000}{-}{4.37216581496386}{×}{{10}}^{{-17}}{}{I}& {0.954308138200000}{-}{1.07137246075397}{×}{{10}}^{{-16}}{}{I}& {0.873643364800000}{+}{1.14626313118273}{×}{{10}}^{{-17}}{}{I}& {0.739663685000000}{+}{7.27845345235421}{×}{{10}}^{{-18}}{}{I}& {0.562212549900000}{-}{4.18686966946647}{×}{{10}}^{{-17}}{}{I}& {0.353862266400000}{+}{8.63971803269712}{×}{{10}}^{{-17}}{}{I}& {0.128973976300000}{-}{3.10136987201386}{×}{{10}}^{{-17}}{}{I}& {-0.0973498701300000}{+}{3.59827545992758}{×}{{10}}^{{-17}}{}{I}& {-0.310339992900000}{+}{1.05230391045198}{×}{{10}}^{{-16}}{}{I}& {-0.496581079300000}{-}{3.11368558291799}{×}{{10}}^{{-17}}{}{I}& {-0.644904545900000}{+}{4.24943082274182}{×}{{10}}^{{-17}}{}{I}& {-0.747090872700000}{+}{3.35757149293384}{×}{{10}}^{{-17}}{}{I}& {-0.798335633500000}{+}{1.32391157403481}{×}{{10}}^{{-18}}{}{I}& {-0.797452935500000}{+}{5.24990660756959}{×}{{10}}^{{-17}}{}{I}& {-0.746810976100000}{+}{1.88718509342198}{×}{{10}}^{{-17}}{}{I}& {-0.652015060100000}{-}{4.10772041263443}{×}{{10}}^{{-17}}{}{I}& {-0.521372067800000}{+}{5.06354772202176}{×}{{10}}^{{-17}}{}{I}& {-0.365185579700000}{-}{2.63851053291697}{×}{{10}}^{{-17}}{}{I}& {-0.194941578200000}{-}{5.03499439915058}{×}{{10}}^{{-17}}{}{I}& {-0.0224501809100001}{+}{4.52079327377909}{×}{{10}}^{{-17}}{}{I}& {0.140990985600000}{-}{3.27249742329407}{×}{{10}}^{{-17}}{}{I}& {0.285351147900000}{+}{3.90748907061433}{×}{{10}}^{{-17}}{}{I}& {0.402487331000000}{+}{8.46953348541619}{×}{{10}}^{{-18}}{}{I}& {0.486593693200000}{+}{9.69808292010440}{×}{{10}}^{{-18}}{}{I}& {0.534456276300000}{-}{8.93242509300831}{×}{{10}}^{{-17}}{}{I}& {0.545506335200000}{-}{2.17628287184369}{×}{{10}}^{{-19}}{}{I}& {0.521681108100000}{+}{2.64823837857832}{×}{{10}}^{{-17}}{}{I}& {0.467114932100000}{+}{7.21282754100757}{×}{{10}}^{{-17}}{}{I}& {0.387694953500000}{+}{8.14606194735691}{×}{{10}}^{{-17}}{}{I}& {0.290523619200000}{-}{3.30222440706408}{×}{{10}}^{{-17}}{}{I}& {0.183334232400000}{+}{1.30397865400380}{×}{{10}}^{{-16}}{}{I}& {0.0739060695099999}{-}{2.50432650283726}{×}{{10}}^{{-17}}{}{I}& {-0.0304778754700000}{+}{1.14628954005974}{×}{{10}}^{{-16}}{}{I}& {-0.123493972100000}{+}{3.96110987542138}{×}{{10}}^{{-17}}{}{I}& {-0.200134182300000}{+}{5.88899487371556}{×}{{10}}^{{-18}}{}{I}& {-0.256938078600000}{+}{2.37401753883706}{×}{{10}}^{{-17}}{}{I}& {-0.292094132000000}{+}{2.45784344714926}{×}{{10}}^{{-17}}{}{I}& {-0.305414195600000}{-}{1.81901620001576}{×}{{10}}^{{-17}}{}{I}& {-0.298194359000000}{-}{5.77214813938006}{×}{{10}}^{{-18}}{}{I}& {-0.272982780500000}{+}{2.58404501063965}{×}{{10}}^{{-18}}{}{I}& {-0.233280250800000}{+}{3.26188112107199}{×}{{10}}^{{-18}}{}{I}& {-0.183201943300000}{+}{4.96078486908024}{×}{{10}}^{{-17}}{}{I}& {-0.127129018800000}{-}{1.29030009412563}{×}{{10}}^{{-16}}{}{I}& {-0.0693767375000000}{-}{1.88682391650801}{×}{{10}}^{{-17}}{}{I}& {-0.0139018249099999}{-}{4.58125235420749}{×}{{10}}^{{-17}}{}{I}& {0.0359334360800001}{-}{4.03182288885964}{×}{{10}}^{{-17}}{}{I}& {0.0775290216300001}{-}{1.02260466177041}{×}{{10}}^{{-16}}{}{I}& {0.109141879100000}{-}{9.03005370913936}{×}{{10}}^{{-17}}{}{I}& {0.129908597500000}{-}{1.03437407935536}{×}{{10}}^{{-16}}{}{I}& {0.139802438800000}{-}{5.48301125806298}{×}{{10}}^{{-17}}{}{I}& {0.139535387900000}{+}{1.70362677499389}{×}{{10}}^{{-17}}{}{I}& {0.130418834600000}{-}{8.18483088226964}{×}{{10}}^{{-17}}{}{I}& {0.114198056900000}{-}{4.62524740296401}{×}{{10}}^{{-17}}{}{I}& {0.0928758766799999}{+}{5.42014083355333}{×}{{10}}^{{-18}}{}{I}& {0.0685398341199999}{-}{1.13457845774918}{×}{{10}}^{{-17}}{}{I}& {0.0432052166399999}{+}{2.44719994645066}{×}{{10}}^{{-17}}{}{I}& {0.0186835400899998}{-}{4.76067169553276}{×}{{10}}^{{-17}}{}{I}& {-0.00351705600300002}{-}{4.90874179457507}{×}{{10}}^{{-18}}{}{I}& {-0.0222562911700000}{+}{2.37790804659256}{×}{{10}}^{{-17}}{}{I}& {-0.0367909819799999}{-}{8.07125429997171}{×}{{10}}^{{-18}}{}{I}& {-0.0467716093300000}{+}{6.56860874183047}{×}{{10}}^{{-18}}{}{I}& {-0.0522109295300000}{+}{4.06939146588404}{×}{{10}}^{{-17}}{}{I}& {-0.0534308314099998}{+}{6.47766995203774}{×}{{10}}^{{-17}}{}{I}& {-0.0509944429499998}{+}{6.45112800847227}{×}{{10}}^{{-17}}{}{I}& {-0.0456306354900000}{+}{1.20150628640086}{×}{{10}}^{{-16}}{}{I}& {-0.0381576393200000}{+}{3.19827732144755}{×}{{10}}^{{-17}}{}{I}& {-0.0294115857600001}{+}{7.32752066151604}{×}{{10}}^{{-17}}{}{I}& {-0.0201845674199999}{-}{2.00493393252301}{×}{{10}}^{{-17}}{}{I}& {-0.0111754014900000}{+}{1.10509346747932}{×}{{10}}^{{-17}}{}{I}& {-0.00295483280499995}{+}{2.44829210702320}{×}{{10}}^{{-17}}{}{I}& {0.00405445532599999}{-}{5.33430418179169}{×}{{10}}^{{-17}}{}{I}& {0.00958384521100003}{+}{4.51135371706707}{×}{{10}}^{{-17}}{}{I}& {0.0135125705899999}{-}{6.60428873666882}{×}{{10}}^{{-17}}{}{I}& {0.0158517096100000}{+}{1.71611595677828}{×}{{10}}^{{-17}}{}{I}& {0.0167211755399999}{+}{6.25557165769816}{×}{{10}}^{{-17}}{}{I}\end{array}\right]$ (11)
 > $\mathrm{verify}\left(\mathrm{data},\mathrm{original},'\mathrm{Array}'\left('\mathrm{float}'\left(10,\mathrm{test}=2\right)\right)\right)$
 ${\mathrm{true}}$ (12)